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Second microlocalization and stabilization of damped wave equations on tori

New challenges in PDE: Deterministic dynamics and randomness in high and infinite dimensional systems October 19, 2015 - October 30, 2015

October 29, 2015 (02:00 PM PDT - 03:00 PM PDT)
Speaker(s): Nicolas Burq (Université de Paris XI)
Location: MSRI: Simons Auditorium
Tags/Keywords
• compact Riemannian manifold

• regularity of initial data

• damping coefficients

• dampened wave equation

• essentially bounded coefficients

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14398

Abstract

We consider the question of stabilization for the damped wave equation on tori
$$(\partial_t^2 -\Delta )u +a(x) \partial _t u =0.$$
When the damping coefficient $a(x)$ is continuous the question is quite well understood and the geometric control condition is necessary and sufficient for uniform (hence exponential) decay to hold. When $a(x)$ is only $L^{\infty}$ there are still gaps in the understanding.
Using second microlocalization we completely solve the question for
Damping coefficients of the form
$$a(x)=\sum_{i=1}^{J} a_j 1_{x\in R_j},$$
Where $R_j$ are cubes.
This is a joint work with P. Gérard

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